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Full text of "Men Of Mathematics"

rnaticians plenty to do in developing their subject freely and
somewhat uncritically  the Euclidean method was for long
neglected in everything but geometry. We have already seen
that the British school applied the method to algebra in the
first half of the nineteenth century. Their successes seem to have
made no very great impression on the work of then* contem-
poraries and immediate successors, and it was only with the
work of Hilbert that the postulational method came to be
recognized as the clearest and most rigorous approach to any
mathematical discipline.
To-day this tendency to abstraction, in which the symbols
and rules of operation in a particular subject are emptied of all
meaning and discussed from a purely formal point of view, is
all the rage, rather to the neglect of applications (practical or
mathematical) which some say are the ultimate human justifi-
cation for any scientific activity. Nevertheless the abstract
method does give insights which looser attacks do not, and in
particular the true simplicity of Boole's algebra of logic is most
easily seen thus.
Accordingly we shall state the postulates for Boolean algebra
(the algebra of logic) and, having done so, see that they can
indeed be given an interpretation consistent with classical logic.
The following set of postulates is taken from a paper by E. V.
Huntington, in the Transactions of the American Mathematical
Society (vol. 85, 1933, pp. 274-304). The whole paper is easily
understandable by anyone who has had a week of algebra, and
may be found in most large public libraries. As Huntington
points out, this first set of his which we transcribe is not as
elegant as some of his others. But as its interpretation in terms
of class inclusion as in formal logic is more immediate than the
like for the others, it is to be preferred here.
The set of postulates is expressed hi terms of -EC, +, X 9 where
K is a class of undefined (wholly arbitrary, without any
assigned meaning or properties beyond those given in the postu-
lates) elements a, 6, c, ... , and a -f b and a x b (written also
simply as ab) are the results of two undefined binary operations,
-f, x ('binary', because each of-f, x operates on two elements
of ). There are ten postulates, I a-VI: